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Understanding Delaunay Triangulations: Algorithms for Optimal Placement

Explore Delaunay triangulation and Voronoi diagrams, algorithms for finding optimal locations, and applications in meshing and Voronoi regions. Learn to compute and utilize circumcenters efficiently. Assigned project to implement Delaunay triangulation with provided graphics. Resources for further reading and homework assignments included.

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Understanding Delaunay Triangulations: Algorithms for Optimal Placement

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  1. Lecture 15: COMPUTATIONAL GEOEMTRY: Delaunay triangulations CS1050: Understanding and Constructing Proofs Spring 2006 Jarek Rossignac

  2. Lecture Objectives Learn the definitions of Delaunay triangulation and Voronoi diagram Develop algorithms for computing them

  3. Find the place furthest from nuclear plants Find the point in the disk that is the furthest from all blue dots

  4. The best place is … The green dot. Find an algorithm for computing it. Teams of 2.

  5. Center of largest disk that fits between points

  6. Algorithm for best place to live max=0; foreach triplets of sites A, B, C { (O,r) = circle circumscribing triangle (A,B,C) found = false; foreach other vertex D {if (||OD||<r) {found=true;};}; if (!found) {if (r>max) {bestO=O; max=r;}; } return (O); Complexity?

  7. Circumcenter pt centerCC (pt A, pt B, pt C) { // circumcenter to triangle (A,B,C) vec AB = A.vecTo(B); float ab2 = dot(AB,AB); vec AC = A.vecTo(C); AC.left(); float ac2 = dot(AC,AC); float d = 2*dot(AB,AC); AB.left(); AB.back(); AB.mul(ac2); AC.mul(ab2); AB.add(AC); AB.div(d); pt X = A.makeCopy(); X.addVec(AB); return(X); }; 2ABAX=ABAB 2ACAX=ACAC AB.left C AC.left AC X A AB B

  8. Delaunay triangles • 3 sites form a Delaunay triangle if their circumscribing circle does not contain any other site.

  9. Inserting points one-by-one

  10. The best place is a Delaunay circumcenter Center of the largest Delaunay circle (stay in convex hull of cites)

  11. Assigned Project • Implement Delaunay triangulation • I provide graphics, GUI, circumcenter • You provide loops and point-in-circle test • Report time cost as a function of input size

  12. Applications of Delaunay triangulations • Find the best place to build your home • Finite element meshing for analysis: nice meshes • Triangulate samples for graphics

  13. School districts and Voronoi regions Each school should serve people for which it is the closest school. Same for the post offices. Given a set of schools, find the Voronoi region that it should serve. Voronoi region of a site = points closest to it than to other sites

  14. Voronoi diagram = dual of Delaunay http://www.cs.cornell.edu/Info/People/chew/Delaunay.html

  15. Assigned Reading • http://www.voronoi.com/applications.htm • http://www.ics.uci.edu/~eppstein/gina/voronoi.html • http://www.cs.berkeley.edu/~jrs/mesh/

  16. Assigned Homework • Definition and algorithms for computing Delaunay triangulation • Duality with Voronoi diagram

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